The Signature of the Chern Coefficients of Local Rings

Abstract

Abstract. This paper considers the following conjecture: If R is an unmixed, equidimensional local ring that is a homomorphic image of a Cohen-Macaulay local ring, then for any ideal J generated by a system of parameters, the Chern coefficient e 1 (J) < 0 is equivalent to R being non Cohen-Macaulay. The conjecture is established if R is a homomorphic image of a Gorenstein ring, and for all universally catenary integral domains containing fields. Criteria for the detection of Cohen-Macaulayness in equi-generated graded modules are derived.

On some conjectures about the Chern numbers of filtrations

“…It is proved for modules over Noetherian local rings in [6]. The next result follows from Proposition 3.2 of [12] and Lemma 2.1 of [3]. We provide another proof.…”

Cohen-Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals

“…Here we should note that Conjecture 1.1 is already solved partially by [5,12]. In fact, Ghezzi, Hong and Vasconcelos [5,Theorem 3.3] proved that the conjecture holds if A is an integral domain which is a homomorphic image of a Cohen-Macaulay ring.…”

“…Here we should note that Conjecture 1.1 is already solved partially by [5,12]. In fact, Ghezzi, Hong and Vasconcelos [5,Theorem 3.3] proved that the conjecture holds if A is an integral domain which is a homomorphic image of a Cohen-Macaulay ring. Mandal, Singh and Verma [12] proved that e 1 (Q) 0 for every parameter ideal Q in an arbitrary Noetherian local ring A and showed that e 1 (Q) < 0, if depth A = d − 1.…”

The Chern Numbers and Euler Characteristics of Modules

“…The authors have examined ( [8], [9], [13], [32]) how the values of e 1 (Q, R) codes for structural information about the ring R itself. More explicitly one defines the set Λ(M ) = {e 1 (Q, M ) | Q is a parameter ideal for M } and examines what its structure expresses about M .…”